Base Mesh changes back to its original form after switching to object mode
Because your object has shapekeys, you have a shapekey selected and therefore you're changing this shapekey, you're not changing the original shape, which is on the top of your list and which is called Basis. If you want to make any change to the original shape, select this shapekey called Basis. If you want to save the shape you've created and make it the basis mesh, go into the dropdown menu on the right (down arrow) and select Move to Top. To make any shapekey appear in Object mode you need to tweak the Value, under the shapekey list.